4.05 Composite functions (Enrichment)
Evaluate the following, given the functions:
\begin{aligned} f \left( x \right) &= 4 x^{3} \\ g \left( x \right) &= x + 5 \end{aligned}Evaluate f \left( g \left( 2 \right) \right), given the functions:
\begin{aligned} f \left( x \right) &= 4 x - 10 \\ g \left( x \right) &= 3 + \dfrac{3}{x}\end{aligned}Consider the following functions:
\begin{aligned} f \left( x \right) = - 2 x - 6 \\ g \left( x \right) = 5 x - 7 \end{aligned}Find f \left( 7 \right).
Hence, evaluate g \left( f \left( 7 \right) \right).
Find g \left( 7 \right).
Hence, evaluate f \left( g \left( 7 \right) \right).
Does f \left( g \left(x\right)\right) = g \left( f \left(x\right)\right) for all x?
Find the composite function f \left( g \left( x \right) \right) for the following:
f \left( x \right) = x^{3} \\ g \left( x \right) = 9 x - 4
f \left( x \right) = x^{2} \\ g \left( x \right) = x^{2} + 1
f \left( x \right) = \sqrt{x} \\ g \left( x \right) = 4 x - 3
f \left( x \right) = 4 x + 9 \\ g \left( x \right) = x^{3}
f \left( x \right) = 4 x - 3 \\ g \left( x \right) = x^{2}
Evaluate the following, given the functions:
\begin{aligned} f \left(x\right) &= - 2 x + 2 \\ g \left(x\right) &= 4 x^{2} - 8 \\ r \left(x\right) &= - 3 x - 8 \end{aligned}g \left( 6 \right)
f \left( g \left( 6 \right) \right)
r\left(f\left(g\left(6\right)\right)\right)
Consider the following functions:
\begin{aligned} f \left( x \right) &= - 2 x - 8 \\ g \left( x \right) &= 4 x^{2} - 4 \end{aligned}Evaluate g \left( f \left( 6 \right) \right).
If h \left( x \right) is defined as f \left( g \left( x \right) \right), state the equation for h \left( x \right).
What type of functions are f \left( g \left( x \right) \right) and g \left( f \left( x \right) \right)?
Consider the following functions:
\begin{aligned} f \left(x\right) &= - 2 x + 6 \\ g \left(x\right) &= 3 x + 1 \end{aligned}If r \left(x\right) is defined as f \left(x^{2}\right), state the equation for r \left(x\right).
Hence, state the equation for q \left(x\right), which is g \left( f \left(x^{2}\right)\right).
Consider the following functions:
\begin{aligned} f \left(x\right) &= x^{2} + 3 \\ g \left(x\right) &= 4 x - 9 \end{aligned}State the equation for f \left( 2 x\right).
Show that f \left( 2 x\right) = g \left( f \left(x\right)\right).
The function f \left(x\right) is defined as f \left(x\right) = - 3 x + 4 g \left(x\right).
Given that f \left(x\right) is a quadratic function, what type of function is g \left(x\right)?
If f \left(x\right) = -3x + 20x^{2}, find the equation for g\left(x\right).
Find an algebraic expression for the function g \left( f \left(x\right)\right).
Consider the following functions:
\begin{aligned} p \left(x\right) &= x + 3 \\ q \left(x\right) &= x^{2} - 1 \\ r \left(x\right) &= \left(x + 4\right) \left(x + 2\right) \end{aligned}Rewrite r \left(x\right) in expanded form.
Show that r \left(x\right) = q \left( p \left(x\right)\right).
Write an algebraic expression for p \left( q \left(x\right)\right).
Find the range of values for which r \left(x\right) < p \left( q \left(x\right)\right).
Consider the following functions:
\begin{aligned} f \left( x \right) &= \dfrac{4}{x} \\ g \left( x \right) &= x^{2} + 3 \end{aligned}Find the following functions:
f \left( x \right) g \left( x \right)
f \left( g \left( x \right) \right)
\dfrac{f \left( x \right)}{g \left( x \right)}
\dfrac{g \left( x \right)}{f \left( x \right)}
Consider the following functions:
\begin{aligned} f \left( x \right) &= 9 x + 4 \\ g \left( x \right) &= x^{2} - 2 x - 3 \end{aligned}Find the following functions:
f \left( x \right) g \left( x \right)
f \left( g \left( x \right) \right)
\dfrac{f \left( x \right)}{g \left( x \right)}
\dfrac{g \left( x \right)}{f \left( x \right)}
For each of the following functions f \left( x \right) and g \left( x \right):
Find the composite function f \left( g \left( x \right) \right).
State the domain of f \left( g \left( x \right) \right).
Find the composite function g \left( f \left( x \right) \right).
State the domain of g \left( f \left( x \right) \right).
\begin{aligned} f \left( x \right) &= \dfrac{9}{1 - 2 x} \\ g \left( x \right) &= \dfrac{3}{x} \end{aligned}
\begin{aligned} f \left( x \right) &= \sqrt{x + 4} \\ g \left( x \right) &= x^{2} - 4 \end{aligned}
Consider the function h \left( x \right) = \sqrt{1 + \sqrt{1 + x}} and suppose that g \left( x \right) = \sqrt{1 + x}.
Find f \left( x \right) such that h(x) = f\left(g\left(x\right)\right).
Consider the function h \left( x \right) = \dfrac{1}{\left(x - 2\right)^{6}}.
If g \left( x \right) = x - 2, find f \left( x \right) such that h(x) = f\left(g\left(x\right)\right).
If f \left( x \right) = \dfrac{1}{x}, find g \left( x \right) such that h(x) = f\left(g\left(x\right)\right).
Consider the function f \left(x\right) = 5 x^{2} + 4. Define g \left(x\right) such that g \left( f \left(x\right)\right) = f \left(x\right) for all x.
For each of the following functions:
State the domain and range of f \left( x \right).
State the domain and range of g \left( x \right).
Hence, determine the domain and range of f \left( g \left( x \right) \right).
\begin{aligned}f \left( x \right) &= x^{2} \\ g \left( x \right) &= x - 7 \end{aligned}
\begin{aligned} f \left( x \right) &= \sqrt{x} \\ g \left( x \right) &= x - 3 \end{aligned}
\begin{aligned} f \left( x \right) &= \sqrt{x} \\ g \left( x \right) &= 4 x + 9 \end{aligned}
\begin{aligned} f \left( x \right) &= x^{3} \\ g \left( x \right) &= x + 8 \end{aligned}
Consider the following functions: \begin{aligned} f \left( x \right) &= \sqrt{x + 5} \\ g \left( x \right) &= 5 x - 6 \end{aligned}
Find an algebraic expression for the function f \left( g \left( x \right) \right).
Express the domain of f \left( g \left( x \right) \right), using inequalities.
Find an algebraic expression for the function g \left( f \left( x \right) \right).
Express the domain of g \left( f \left( x \right) \right), using inequalities.
Consider the following functions:
\begin{aligned} f \left( x \right) &= - 2 x \\ g \left( x \right) &= \dfrac{x}{x - 5} \end{aligned}Find an algebraic expression for the function f \left( g \left( x \right) \right).
Determine the value(s) of x that are not in the domain of f \left( g \left( x \right) \right).
Find an algebraic expression for the function g \left( f \left( x \right) \right).
Determine the value(s) of x that are not in the domain of g \left( f \left( x \right) \right).
At sale time in a certain clothing store, all dresses are on sale for \$5 less than 75\% of the original price.
Write a function g\left(x\right) that finds 75\% of x.
Write a function f \left(x \right) that finds 5 less than x.
Construct the composite function f \left( g \left( x \right) \right).
Hence, calculate the sale price of a dress that has an original price of \$94.
A computer manufacturer sells hard drives to a retail outlet, each at a cost of \$11 more than the manufacturing cost. The retail store then sells each hard drive to the public, charging 60\% more than they paid to the manufacturer.
If m represents the manufacturing cost (in dollars), find a function A \left( m \right) which returns the price of buying a hard drive at the retail store.
Find the price of buying a hard drive at the store, if the manufacturing cost is \$22.50 .